Shapiro–Francia test compared to other normality test using expected p-value

The Shapiro–Francia (SF) normality test is an important test in statistical modelling. However, little has been done by researchers to compare the performance of this test to other normality tests. This paper therefore measures the performance of the SF and other normality tests by studying the distribution of their p-values. For the purpose of this study, we selected eight well-known normality tests to compare with the SF test: (i) Kolmogorov–Smirnov (KS), (ii) Anderson–Darling (AD), (iii) Cramer von Mises (CM), (iv) Lilliefors (LF), (v) Shapiro–Wilk (SW), (vi) Pearson chi-square (PC), (vii) Jarque– Bera (JB) and (viii) D'Agostino (DA). The distribution of p-values of these normality tests were obtained by generating data from normal distribution and well-known symmetric non-normal distribution at various sample sizes (small, medium and large). Our simulation results showed that the SF normality test was the best test statistic in detecting deviation from normality among the nine tests considered at all sample sizes.     

Randomized Response Techniques Using Maximum Likelihood Estimator

The randomized response (RR) technique with two decks of cards proposed by Odumade and Singh (2009) can always be made more efficient than the RR techniques proposed by Warner (1965), Mangat and Singh (1990), and Mangat (1994) by adjusting the proportion of cards in the decks. Arnab et al. (2012) generalized Odumade and Singh strategy (2009) for complex survey designs and wider class of estimators. In this paper improvement of Arnab et al. (2012) estimator has been made by using maximum likelihood method.     

Optimal two treatment repeated measurement designs for three periods

Repeated Measurement Designs, with two treatments, n (experimental) units and p periods are examined, the two treatments are denoted A and B. The model with independent observations within and between treatment sequences is used. Optimal designs are derived for: (i) the difference of direct treatment effects and the difference of residual effects, (ii) the difference of direct treatment effects, and (iii) the difference of residual effects. We prove that for three periods when n is odd the optimal design in the three cases (i), (ii), and (iii) is determined by taking the sequences BAA and ABB in numbers differing by one. If n is even, the optimal design in cases (i), (ii), and (iii) is again the same, by taking the sequences ABB and BAA in equal numbers. In case (i), for n even or odd, in the optimal design there is no correlation between the two estimated parameters. For n even, case (i) was solved by Cheng and Wu in 1980. The above imply that with two treatments in practice are preferable to use three periods instead of two.     

Modelling Correlated Zero-inflated Count Data

This paper extends the two-component approach to modelling count data with extra zeros, considered by Mullahy (1986), Heilbron (1994) and Welsh et al. (1996), to take account of possible serial dependence between repeated observations. Generalized estimating equations (Liang & Zeger, 1986) are constructed for each component of the model by incorporating correlation matrices into each of the maximum likelihood estimating equations. The proposed method is demonstrated on weekly counts of Noisy Friarbirds ( Philemon cornic-ulatus ), which were recorded by observers for the Canberra Garden Bird Survey (Hermes, 1981).     

Some Applications of Order Statistics to Multivariate Analysis II

ABSTRACT

Harter (1979) summarized applications of order statistics to multivariate analysis up through 1949. The present paper covers the period 1950–1959. References in the two papers were selected from the first and second volumes, respectively, of the author's chronological annotated bibliography on order statistics [Harter (1978, 1983)]. Tintner (1950a) established formal relations between four special types of multivariate analysis: (1) canonical correlation, (2) principal components, (3) weighted regression, and (4) discriminant analysis, all of which depend on ordered roots of determinantal equations. During the decade 1950–1959, numerous authors contributed to distribution theory and/or computational methods for ordered roots and their applications to multivariate analysis. Test criteria for (i) multivariate analysis of variance, (ii) comparison of variance–covariance matrices, and (iii) multiple independence of groups of variates when the parent population is multivariate normal were usually derived from the likelihood ratio principle until S. N. Roy (1953) formulated the union–intersection principles on which Roy & Bose (1953) based their simultaneous test and confidence procedure. Roy & Bargmann (1958) used an alternative procedure, called the step–down procedure, in deriving a test for problem (iii), and J. Roy (1958) applied the step–down procedure to problem (i) and (ii), Various authors developed and applied distribution theory for several multivariate distributions. Advances were also made on multivariate tolerance regions [Fraser & Wormleighton (1951), Fraser (1951, 1953), Fraser & Guttman (1956), Kemperman (1956), and Somerville (1958)], a criterion for rejection of multivariate outliers [Kudô (1957)], and linear estimators, from censored samples, of parameters of multivariate normal populations [Watterson (1958, 1959)]. Textbooks on multivariate analysis were published by Kendall (1957) and Anderson (1958), as well as a monograph by Roy (1957) and a book of tables by Pillai (1957).     

CVaR-EVT和BMM在极端金融风险管理中的应用研究

随着风险度量一致性原则的提出,研究发现金融机构广泛采用的VaR模型存在严重不足,尤其针对分布具有厚尾特征的极端金融风险无法有效度量。本文采用极值理论（EVT）解决VaR方法的尾部度量不足问题,利用CVaR-EVT和BMM模型分析美国、香港股票市场和我国沪深两市指数18年的日收益数据,研究发现：（1）在95％置信区间及点估计中,分位数为99%的CVaR-EVT所揭示的极端风险优于VaR的估计值;且BMM方法为实施长期极端风险管理提供了有力决策依据,其回报率受分段时区的影响,期间越长,风险估计值越高;（2）模型采用ML和BS方法统计估值显示,我国股票市场极端风险尾部估计值高于香港和美国市场;但是,国内市场逐步稳定,并呈现出跟进国际市场且差距缩小的发展趋势。     

Another look at plotting positions

Probability paper was used as early as 1896, and was mentioned in the literature more than 30 times before 1950, mainly by hydrologists, most of whom used the plotting position (i-0.5)/n proposed by Hazen (1914). Gumbel (1942a) considered the modal position (i-1)/(n-1) and the mean position i/(n+1) [the latter proposed by Weibull (1939a,b)], and chose the latter. Lebedev (1952) and others proposed the use of (i-0.3)/(n+0.4), which is approximately the median position advocated by Johnson (1951). Blom (1958) sug-gested (i-α)/(n-2α+1), where a is a constant (usually 0 ≤ α ≤ 1), which includes all of the above plotting positions as special cases. Moreover, by proper choice of α, one can approximate F[E(x_i)], the position proposed by Kimball (1946), for any distri-bution of interest. Gumbel (1954) stated five postulates which plotting positions should satisfy. Chernoff & Lieberman (1954) discussed the optimum choice of plotting positions in various situ-ations. It is clear that the optimum plotting position depends on the use that is to be made of the results and may also depend on the underlying distribution. The author endeavors to formulate recommendations as to the best choice in various situations.     

Autoregression quantiles and related rank score processes for generalized random coefficient autoregressive processes

Regression quantiles were developed by Koenker and Bassett (Econometrica 46 (1978), 33–50); they provide natural and extremely useful counterparts of the sample quantiles in general linear models. The regression rank scores were introduced by Gutenbrunner and Jurečková (Ann. Statist. 8 (1992), 305–329) as dual variables to regression quantiles. Koul and Saleh (Ann. Statist. 23 (1995), 670–689) developed the procedures based on the regression quantiles of Koenker and Bassett (Econometrica 46 (1978), 33–50) and the regression rank scores of Gutenbrunner and Jurečková Ann. Statist. 8 (1992), 305–329 in linear regression to the pth-order autoregression models. In this paper, we further develop and investigate the analogs of these procedures to a larger class of processes and derive a test for a bilinear model without estimating the bilinear coefficient and the autoregression constants.     

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.     

On-line control procedures for integrated moving average process of order one

A commonly used model in process adjustment is the integrated moving average (IMA) process of order one. When the disturbances are normally distributed, this problem has been considered by Box and Jenkins (1963, 1976), Kramer (1989), Srivastava and Wu (1992), Box and Kramer (1992) and Srivastava (1996). Box and Lucefio (1994) extended the above model to any disturbance that is symmetrically distributed. However, their solution involves Fred holm integral equation which must be solved by numerical methods. In this paper, we follow the approach of Srivastava and Wu (1992, 1996) and provide the optimum control limit explicitly in terms of the first moment of the randomly stopped random sum of independent and identically distributed observations. These results are also used to study the robustness of the normal model. It is shown that the procedure is robust for small to moderate departure from normality.     

THE EFFICIENCY OF TIME SERIES ESTIMATORS FOR REPEATED SURVEYS

The classical approach to the analysis of data from repeated surveys, based on Patterson (1950), has recently been extended by the work of Blight and Scott (1973) and Scott and Smith (1974), by assuming that a time series relationship exists between the population parameters at different times. The purpose of this paper is to compare the efficiencies of these three approaches by computing the mean square error (MSE) of the estimators of the current mean and of the change in mean on the last two occasions.     

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matr ix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.     

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg’s (1990) elegant graph-theoretic characterization of the Markov equivalence of chain graphs.     

A comparison of count data models with an application to daily cigarette consumption of young persons

The objective of this study is providing a comparative assessment for researchers to deal with the challenges of analyzing count data and examining the factors associated with daily cigarette consumption among the young people in Turkey. We fitted Poisson (P), negative binomial (NB), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), Poisson hurdle (PH) and negative binomial hurdle (NBH) regressions to cigarette consumption count data by using the 2014 Turkey Health Survey. Our results showed that the ZINB and NBH models should be preferred. We also found that, gender, employment and tobacco use at home are more effective factors for smokers and nonsmokers in the 15–24 age group in Turkey.     

Multi-step quantile regression tree

Quantile regression (QR) proposed by Koenker and Bassett [Regression quantiles, Econometrica 46(1) (1978), pp. 33–50] is a statistical technique that estimates conditional quantiles. It has been widely studied and applied to economics. Meinshausen [Quantile regression forests, J. Mach. Learn. Res. 7 (2006), pp. 983–999] proposed quantile regression forests (QRF), a non-parametric way based on random forest. QRF performs well in terms of prediction accuracy, but it struggles with noisy data sets. This motivates us to propose a multi-step QR tree method using GUIDE (Generalized, Unbiased, Interaction Detection and Estimation) made by Loh [Regression trees with unbiased variable selection and interaction detection, Statist. Sinica 12 (2002), pp. 361–386]. Our simulation study shows that the multi-step QR tree performs better than a single tree or QRF especially when it deals with data sets having many irrelevant variables.     

Some Related Minima Stability and Minima Infinite Divisibility of the General Multivariate Pareto Distributions

This article studies the minima stable property of the general multivariate Pareto distributions MP^(k)(I), MP^(k)(II), MP^(k)(III), MP^(k)(IV) which can be applied to characterize the MP^(k) distribution via its weighted ordered coordinates minima and marginal distribution. Also, the multivariate semi-Pareto distribution (denoted by MSP) is discerned in the class of geometric minima infinite divisible and geometric minima stable distributions. If the exponent measure is satisfied by some functional equation, then the geometric minima stable property can be used to characterize the MSP distribution. Finally, the finite sample minima infinite divisible property of the MP^(k)(I), (II), and (IV) distributions is also discussed.     

Testing for superadditivity of the mean value function of a non-homogeneous poisson process

Consider a non-homogeneous Poisson process, N(t), with mean value function Λ(t) and intensity function λ(t). A conditional test of the hypothesis that the process is homogeneous, versus alternatives for which Λ(t) is superadditive, was proposed by Hollander and Proschan (1974). A new test for superadditivity of Λ(t), which is based on a linear combination of the occurrence times of the process N{t) is suggested in this paper. Though this test has the same Pitman efficiency as the Hollander-Proschan test, it is shown by Monte-Carlo simulation that our test has more power for many important alternatives. Tables for the exact null distribution of the test statistic have been given.     

An integral representation technique for calculating general multivariate probabilities with an application to multivariate x2

Recently in Dutt (1973, (1975), intgral representations over (0,A) were obtained for upper and lover multivariate normal and the probilities. It was pointed out that these integral representaitons when evaluated by Gauss-Hermite uadrature yield rapid and accurate numerical results.

Here integral representaitons, based on an integral formula due to Gurland (1948), are indicated for arbitrary multivariate probabilities. Application of this general representaion for computing multivariate x2 probabilities is discussed and numerical results using Gaussian quadrature are given for the bivariate and equicorre lated trivariate cases. Applications to the multivariate densities studied by Miller (1965) are also included     

Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.     

A new three-parameter ageing distribution

We propose a new three-parameter ageing distribution called the Weibull-Poisson (WP) distribution, which generalizes the exponential-Poisson (EP) distribution introduced by Kus (2007). This new distribution has a more general form of failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model three ageing classes of life distributions including decreasing failure rate (DFR), increasing failure rate (IFR), and modified upside-down-bathtub (MUBT)-shaped failure rate. It thus provides an alternative to many existing life distributions. Various properties of this distribution are discussed and the estimation of the parameters is considered by the expectation maximization (EM) algorithm. Also, the asymptotic variance-covariance matrices of these estimates are obtained. Furthermore, some expressions for the Rènyi and Shannon entropies are given. Simulation studies are performed and experimental results are illustrated based on a real data set.